Short rank-metric codes and scattered subspaces

By exploiting the connection between scattered ${\mathbb{F}}_q$-subspaces of ${\mathbb{F}}_{q^m}^3$ and minimal non degenerate $3$-dimensional rank metric codes of ${\mathbb{F}}_{q^m}^n$, $n\ge m+2$, described in [2], we will exhibit a new class of codes with parameters $[m+2,3,m-2{]}_{q^m\mathrm{/}q}$ for infinite values of $q$ and $m\ge 5$ odd. Moreover, by studying the geometric structures of these scattered subspaces, we determine the rank weight distribution of the associated codes.